High-contrast homogenization: Entry.Response.Rows.RecentredResponseEnergyBound #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
The centred response as a sheared block energy, and its three-row bound #
The response energy of a block under a shear congruence equals the response energy of the original
block at the correspondingly transformed load; specializing this to the recentring shear
G_g = ((1,0),(g,1)) expresses the two centred responses Jtilde^-(e) and Jtilde^+(e) of
e.response.cutoff.estimate, and their adjoint recentred blocks Ehat_t^∓, in swapped and closed
forms, and the adjoint annealed energies of the cutoff rows follow in the same closed block form.
Chaining these identifications expresses each centred response as the annealed energy of the
recentred block at the shifted load minus half the pairing of the two coordinates of the annealed
mean Y^∓. The cutoff decomposition of the cutoff pairing exhibits that half-pairing itself as the
cutoff half-energy minus the two cutoff-mean pairings plus half the self-pairing of Y^∓, with the
cutoff pairing row bounded by the expected absolute pairing (AK.HC (3.45)-(3.54)); combining the two
identifications gives the three-row bound on |Jtilde^∓(e)|, for both signs.
The adjoint annealed energies of the cutoff rows in closed block form #
The adjoint cutoff rows of the response estimate bound the centred response with the two error
scalars respEJPlus and respTauPlus. The first is the P-expectation of the adjoint recentred
pathwise response at one adapted cell; the second is the defect of that expectation between two
scales. Under the entrywise integrability of the coarse block carried by the cutoff rows, both are
algebraic functions of the adjoint recentred annealed block respEhatPlus: the pathwise response
of a₊ = aᵗ + g is the block energy x · A x / 2 - p · q', so its expectation is the same block
energy of the annealed block E[A], and the pairing p · q' cancels in the defect. These are the
closed block forms against which the adjoint cutoff rows are stated (e.response.cutoff.estimate).
The adjoint annealed energy of the cutoff rows as a block quadratic. Under entrywise
integrability of the coarse block, the P-expectation of the adjoint recentred pathwise response
J(U_t, p, q⁺; a₊) is the quadratic form ½ x · E[A] x of the adjoint recentred annealed block
respEhatPlus at the load x = (-p, q⁺), minus the pairing p · q⁺. This is the closed block
form of the centred adjoint response energy e.response.cutoff.estimate.
The shear form of the annealed energies of the cutoff rows #
The cutoff rows of the response estimate bound the centred response Jtilde^±(e) by expressions
in the expectation of the recentred pathwise response, its scale defect, and the source load
(e.response.cutoff.estimate, AK.HC (3.45)–(3.54)). The centred response itself is defined on the
annealed block E_t of the original coefficient, while the two energy scalars are the block
energies of the recentred blocks Ehat_t^∓. The two objects are compared only after the algebraic
step recorded here.
The recentring a_∓ = a ∓ g is the constant shear G^t A G of the doubled block, and the shear
moves the flux load: the block response energy of the shear congruence at a load (p, r) is the
response energy of the original block at the load (p, r - g p). Since g is skew, the pairing
p. g p vanishes, so the constant p. q' of the energy is unchanged and the identity is exact.
This is the algebraic half of the recentring of Step 4 of the response argument, and the first step
that puts the recentred block energies on the same footing as the centred response.
The shear congruence shifts the flux load in the block response energy. For the constant
shear G = ((1, 0), (g, 1)) with g skew and any doubled block A, the response energy of the
shear congruence at the load (p, r) is the response energy of A at the load
(p, r - g p):
blockResponseEnergy (Gᵀ A G) p r = blockResponseEnergy A p (r - matVecMul g p).
The constant p. r of the energy is unchanged because p. (g p) = 0. This is the algebraic
content of the recentring of the coefficient in Step 4 of the response argument
(e.response.cutoff.estimate).
The centred response under the recentring shear #
The recentring of the coefficient by a constant skew matrix g is the shear congruence Gᵀ A G
of the doubled block with G = ((1, 0), (g, 1)). The block response energy of the shear
congruence at the load (p, r) is the response energy of the original block at the load
(p, r - g p), and the constant p. r of that energy is unchanged because p. (g p) = 0.
This file records the corresponding statements for the doubled block-vector multiplication and for
the two terms of the centred response: the block mean of the shear congruence is the block mean of
the original block at the shifted load, with its two slots related by Y₂ ↦ Y₂ - g Y₁, so the
centring pairing is shear-invariant because Y₁. (g Y₁) = 0. The centred response therefore
transforms exactly as the block response energy does, which puts the centred responses on the
recentred blocks of e.response.cutoff.estimate.
The shear congruence Gᵀ A G with G = ((1, 0), (g, 1)) acts on a doubled vector by the
triangular change of variables (X₁, X₂) ↦ (X₁, g X₁ + X₂) in the second slot, and then adds the
first slot gᵀ times the second slot of the image: the first component of the image is
(A (X₁, g X₁ + X₂))₁ + gᵀ (A (X₁, g X₁ + X₂))₂ and the second component is
(A (X₁, g X₁ + X₂))₂.
The centring pairing of the block mean is invariant under the shear congruence
G = ((1, 0), (g, 1)) with g skew. Writing Z = A (-p, r - g p), the mean of the shear
congruence at the load (-p, r) has slots (-p + Z₂, r + Z₁ + gᵀ Z₂) while the mean of A at the
shifted load (-p, r - g p) has slots (-p + Z₂, r - g p + Z₁), and the second slots differ by
-g (-p + Z₂), whose pairing against -p + Z₂ vanishes because g is skew. This is the
centring half of the recentring of Step 4 of the response argument
(e.response.cutoff.estimate).
The centred response is invariant under the shear recentring:
blockCenteredResponse (Gᵀ A G) p r = blockCenteredResponse A p (r - g p) for the constant shear
G = ((1, 0), (g, 1)) with g skew. The energy term transforms by
blockResponseEnergy_blockCongr_shear and the centring pairing by shear invariance, so the centred
responses on the recentred blocks and on the original blocks are the same scalar. This is the
passage from the centred responses to the annealed energies of
e.response.cutoff.estimate.
The centred responses on the recentred blocks, in swap form #
The two centred responses Jtilde^- and Jtilde^+ of e.response.cutoff.estimate are defined
on the annealed block E_t = respMean at the skew-shifted loads. The recentring shear
G_g = ((1, 0), (g, 1)) of Step 4 carries that block to Ehat_t^- = G_gᵀ E_t G_g, and the
adjoint recentring carries the adjoint block to Ehat_t^+ = G_{-g}ᵀ (D E_t D) G_{-g}. Under the
shear, the centred response is unchanged once the second load is shifted back by g p, so each
Jtilde^± is the centred response of the recentred block Ehat_t^± at the recentred load
q^±. Because the reciprocal normalizations give p. q^± = |e|^2 = 1, the swap form of the
centred response then collapses each Jtilde^± to the closed expression
-1/2 - (1/4) <Ehat_t^± x^±, R Ehat_t^± x^±>. Thus the centred response measures exactly the
failure of the recentred block to be reciprocal along its own load, and it vanishes precisely
when (1/4) <Ehat x, R Ehat x> = -1/2.
The centred response Jtilde^-(e) of e.response.cutoff.estimate is the centred response
of the recentred block Ehat_t^- at the recentred load q^-: the shear congruence
Ehat_t^- = G_gᵀ E_t G_g acts on the doubled load by the triangular change of variables
(X₁, X₂) ↦ (X₁, g X₁ + X₂), and since the centred response is shear-invariant the definition
of respCenteredJMinus on E_t at (p, q - h_t p) agrees with the response of Ehat_t^- at
(p, q^-) with q^- = q + (g - h_t) p.
The adjoint centred response Jtilde^+(e) of e.response.cutoff.estimate is the centred
response of the recentred adjoint block Ehat_t^+ at the recentred load q^+. The adjoint
recentred block is the (-g)-shear congruence Ehat_t^+ = G_{-g}ᵀ (D E_t D) G_{-g}, and the
centred response of that congruence at (p, q^+) is the response of the adjoint block at
(p, q^+ + g p), which is the definition of respCenteredJPlus on the adjoint annealed block
at (p, q + h_t p).
The centred response as the annealed energy minus half the mean pairing #
The centred responses Jtilde^-(e) and Jtilde^+(e) of e.response.cutoff.estimate are
defined on the annealed block E_t = respMean at the skew-shifted loads. The recentring shear
G = respG F carries that block to the recentred block Ehat_t^- = respEhatMinus, and the
adjoint recentring carries the adjoint block to Ehat_t^+ = respEhatPlus. Under that
identification the energy part of the centred response is the P-expectation of the recentred
pathwise response, and its remaining term is half the pairing of the two coordinates of the
annealed mean Y^± of AK.HC (2.32). The two identities below record the resulting closed form
of the centred response as the annealed energy E[J_t^±] minus that pairing, which is the form
in which the cutoff decomposition of e.response.cutoff.estimate consumes it.
The centred response Jtilde^-(e) of e.response.cutoff.estimate is the annealed energy
E[J_t^-] = respEJMinus of the recentred coefficient a_- = a - g minus half the pairing of
the two coordinates of the annealed mean Y^- = (I_{2d} + R Ehat_t^-) x^- of AK.HC (2.32),
with x^- = (-p, q^-) the doubled recentred load. The energy part is the block quadratic of the
recentred annealed block Ehat_t^- = respEhatMinus at that load, so the two sides agree
definitionally once the annealed energy is put in closed block form.
The adjoint centred response Jtilde^+(e) of e.response.cutoff.estimate is the adjoint
annealed energy E[J_t^+] = respEJPlus of the recentred coefficient a_+ = a^t + g minus half
the pairing of the two coordinates of the annealed mean Y^+ = (I_{2d} + R Ehat_t^+) x^+ of
AK.HC (2.32), with x^+ = (-p, q^+) the doubled recentred load. This is the adjoint twin of
respCenteredJMinus_eq_respEJMinus_sub, and the two sides agree definitionally once the adjoint
annealed energy is put in closed block form.
The cutoff pairing row against the absolute pairing (AK.HC (3.45)-(3.54)) #
The half-pairing (1 / 2) * cutoffPairingOnCellAux is the cutoff pairing row of the centred
decomposition of e.response.cutoff.estimate. Its expectation is bounded by one half of the
expected absolute pairing, and a fortiori by the whole expected absolute pairing, which is the
last summand carried by the cutoff estimate.
Neither bound needs integrability: when the pairing is not integrable the Bochner integral takes
the junk value zero, so the inequalities read 0 ≤ 0 and 0 ≤ a nonnegative integral.
The cutoff pairing row (1 / 2) * cutoffPairingOnCellAux is bounded in absolute value by
half of the expected absolute pairing. No integrability of the pairing is assumed: both sides are
read through the Bochner integral.
The centred three-row form of the integrated cutoff decomposition (AK.HC (3.45)-(3.54)) #
The cutoff pairing at a sample coefficient expands pathwise into the cutoff half-energy, the two
cutoff-mean pairings and half the self-pairing of the annealed mean Y. Integrating that
expansion against the law P and rearranging gives the identity
∫J - (1/2) Y₁ · Y₂ = ∫A - ∫(E - J) + (1/2) <∫M₁ - Y₁, Y₂> + (1/2) <Y₁, ∫M₂ - Y₂>,
whose three rows are the cutoff pairing, the cutoff energy defect and the CENTRED cutoff-mean
row. Centring the mean row is the mean cancellation of AK.HC (3.45)-(3.54): when the cutoff has
cell average one and the inserted loads are the actual annealed means, ∫M - Y is the annealed
(φ - 1)-weighted mean and is small, whereas the uncentred row is not.
The integral of vecDot (M a) Y over P is the dot product of Y with the vector of
coordinate integrals of M: the Fubini step identifying one mean pairing of the integrated
cutoff decomposition of e.response.cutoff.estimate (AK.HC (3.45)-(3.54)).
The centred three-row integrated cutoff decomposition (e.response.cutoff.estimate,
AK.HC (3.45)-(3.54)). If the half-pairing A is pathwise the affine combination
E - (1/2) M₁ · Y₂ - (1/2) Y₁ · M₂ + (1/2) Y₁ · Y₂ of the half-energy E, the cutoff-mean
coordinates M₁, M₂, the annealed mean Y and the constant (1/2) Y₁ · Y₂, then integrating
against the probability law P and identifying the integrated mean pairings by Fubini bounds the
centred response ∫J - (1/2) Y₁ · Y₂ by the absolute cutoff pairing |∫A|, the absolute cutoff
energy defect |∫(E - J)| and the CENTRED cutoff-mean row
(1/2) |<∫M₁ - Y₁, Y₂> + <Y₁, ∫M₂ - Y₂>|.
The centred response against its three cutoff rows (AK.HC (3.45)-(3.54)) #
The centred responses Jtilde^±(e) of e.response.cutoff.estimate are the annealed energies at
the recentred loads minus half the pairing of the two coordinates of the annealed mean Y^±. The
cutoff decomposition of the cutoff pairing exhibits the half-pairing as the cutoff half-energy
minus the two cutoff-mean pairings plus half the self-pairing of Y^±, so integrating against the
law and bounding the result reduces the centred response to three rows: the expected absolute
cutoff pairing, the absolute cutoff energy defect E[cutoffHalfEnergy - J], and the CENTRED
cutoff-mean row, in which the integrated cutoff means appear only through their difference from the
annealed mean. These are the e4, e1 and e2 + e3 of the printed variational calculation.
The centred response against its three cutoff rows (e.response.cutoff.estimate,
AK.HC (3.45)-(3.54)), minus carriers. If the half-pairing
(1/2) * cutoffPairingOnCellAux is pathwise the affine combination of the cutoff half-energy,
the two cutoff-mean coordinates and half the self-pairing of the annealed mean Y^-, then the
centred response |Jtilde^-(e)| is bounded by the sum of the expected absolute cutoff pairing, the
absolute cutoff energy defect E[cutoffHalfEnergy - J], and the CENTRED cutoff-mean row
(1/2) |<E[M₁] - Y₁^-, Y₂^-> + <Y₁^-, E[M₂] - Y₂^->|.
The centred response against its three cutoff rows (e.response.cutoff.estimate,
AK.HC (3.45)-(3.54)), plus carriers. If the half-pairing
(1/2) * cutoffPairingOnCellAux is pathwise the affine combination of the cutoff half-energy,
the two cutoff-mean coordinates and half the self-pairing of the annealed mean Y^+, then the
centred response |Jtilde^+(e)| is bounded by the sum of the expected absolute cutoff pairing, the
absolute cutoff energy defect E[cutoffHalfEnergy - J], and the CENTRED cutoff-mean row
(1/2) |<E[M₁] - Y₁^+, Y₂^+> + <Y₁^+, E[M₂] - Y₂^+>|.